Method and apparatus for controlling waveguide birefringence by selection of a waveguide core width for a top cladding

ABSTRACT

A method and apparatus for controlling waveguide birefringence by selection of a waveguide core width for a tuned top clad is described herein. In one example, a dopant concentration within a top cladding material is between 3-6% (wt.). Given a tuned top cladding composition, a width of the waveguide core is pre-selected such that birefringence is minimized, i.e., a zero, or near zero. The desirable width of the waveguide core is determined by calculating the distribution of stress in the top cladding over a change in temperature. From this distribution of stress, a relationship between the polarization dependent wavelength and variable widths of the waveguide in the arrayed waveguide grating are determined. This relationship determines a zero value, or near zero value, of polarization dependent wavelength for a given range of waveguide widths. Accordingly, the width of the waveguide may be selected such that the polarization dependent wavelength is minimized.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of copending PCT applicationPCT/US2005/036628 designating the U.S. to Parhami et al., filed on Oct.11, 2005, entitled “Method and Apparatus for Controlling WaveguideBirefringence by Selection of a Waveguide Core Width for a TopCladding,” incorporated herein by reference.

TECHNICAL FIELD

The present invention relates generally to methods and apparatus forcontrolling waveguide birefringence. More particularly, the presentinvention relates to controlling waveguide birefringence by theselection of a waveguide core width for a given top cladding.

BACKGROUND

The increase in internet traffic and other telecommunications over thepast several years has caused researchers to explore new ways toincrease fiber optic network capacity by carrying multiple data signalsconcurrently through telecommunications lines. To expand fiber networkcapacity, fairly complex optical components have already been developedfor wavelength division multiplexing (WDM) and dense wavelength divisionmultiplexing (DWDM).

Planar lightwave circuit (PLC) technology is one technology that may beused to implement optical wavelength routers. In a PLC, light isrestricted to propagate in a region that is thin in two dimensions,referred to herein as the transverse dimension and the lateraldimension, and extended in the other dimension, hi a conventional PLC, acore layer typically lies between a top cladding layer and a bottomcladding layer and channel waveguides are often formed by at leastpartially removing (typically through an etching process) core materialbeyond the transverse limits of the channel waveguide and replacing itwith at least one layer of side cladding material that has an index ofrefraction that is lower than that of the core material. The sidecladding material is usually the same material as the top claddingmaterial.

Each layer is typically doped in a manner such that the core layer has ahigher index of refraction than either the top cladding or bottomcladding. When layers of silica glass are used for the optical layers,the layers are typically deposited on a silicon wafer. Depositionprocesses include chemical vapor deposition (CVD), low pressure chemicalvapor deposition (LPCVD), and/or plasma-enhanced CVD (PECVD). Moreover,one or more of the optical layers of the slab waveguide and/or channelwaveguide may comprise an optically transparent polymer. For example,spin coating is one known film deposition method. A doped-silicawaveguide is usually preferred because it has a number of attractiveproperties including low cost, low loss, low birefringence, stability,and compatibility for coupling to fiber.

The arrayed-waveguide grating router (AWGR) is an example of anintegrated optical router. An AWGR is a PLC having at least one inputchannel waveguide, an input planar waveguide, an arrayed-waveguidegrating (AWG), an output planar waveguide, and at least one outputchannel waveguide. Alternatively, an AWGR may comprise a plurality ofoutput waveguides and a plurality of input waveguides. AWGRs may beconfigured to perform a variety of functions, for instance, they mayfunction as multiplexers, demultiplexers, or they may be configured toperform both functions.

One aspect of performance that is affected by the present invention isreferred to as polarization dependent wavelength (PDW). This term, aswell as a number of related terms, will now be defined. Spectraltransmissivity (in units of dB) is defined as the optical power (inunits of dBm) of substantially monochromatic light that emerges from thefiber that is coupled to the input port minus the optical power (inunits of dBm) of the light that enters the optical fiber that is coupledto the output port of the optical router. Spectral transmissivity is afunction of the selected input port, the selected output port, theoptical wavelength, and the polarization state of the incident light.When the incident light is in a polarization state called a “principalstate of polarization,” the light will be in the same polarization statewhen it emerges from the device. For purposes of illustration only, theprinciple states of polarization are assumed to be independent ofwavelength, input port and output port. It is understood that theinvention is not so limited by this assumption.

Again, for the purposes of illustration only, it will be assumed thatthe two principal states of polarization are the so-called transverseelectric (TE) and transverse magnetic (TM) polarization states. The TEpolarization state has an electric field that is predominantly alignedin the transverse direction and the TM polarization state has anelectric field that is predominantly aligned in the lateral direction.Again, the invention is not so limited to devices having these principlestates of polarization.

Typically, for values of spectral transmissivity that are larger than−10 dB, the TM spectral transmissivity is a replica of the TE spectraltransmissivity that is shifted in wavelength by an amount that isreferred to as the polarization dependent dispersion (PDD). PDD ispositive if the TM spectral transmissivity has a maximum that has alonger wavelength than the maximum of the TE spectral transmissivity andis negative otherwise. PDW is defined as the absolute value of the PDD.

In many fiber optic communication systems, the polarization state of thelight in the optical fiber may change in a manner that is uncontrolledand unpredictable. A change in the polarization state of the light inthe fiber as it enters an AWG will cause a change in the optical powerthat emerges from the AWG that may be as large as the value ofpolarization dependent loss (PDL) for the AWG. There have been a numberof techniques developed in an attempt to minimize PDW.

One approach to minimizing PDW involves selecting an optical layerdesign with minimum birefringence. In one example of this approach, U.S.Pat. No. 5,930,439 (Ojha et al.), which is incorporated herein byreference in its entirety, discloses a planar optical waveguide whichreduces birefringence by doping the various optical layers so that thetop cladding has a thermal coefficient of expansion that is close to thethermal expansion coefficient of the substrate.

Another approach is described in A. Kilian et al., Birefringence FreePlanar Optical Waveguide Made by Flame Hydrolysis Deposition (FHD)Through Tailoring of the Overcladding, Journal of Lightwave Technology,v. 18, no. 2, p. 193 (2000), which is incorporated herein by referencein its entirety. Kilian discloses that because the thermal expansion ofthe top cladding largely determines the birefringence in the waveguide,a top cladding can be developed and made with aflame-hydrolysis-deposition (FHD) process to reduce the birefringence.

Many methods and apparatus attempt to formulate a top cladding materialwhich reduces birefringence. However, this may require specificcompositions for the top cladding. These techniques, as well as others,teach the application of a specific top cladding composition to reducebirefringence. However, there is a need for a reliable method to reducethe birefringence in a waveguide for varying top cladding compositions.

SUMMARY

The top cladding material which usually covers the waveguides andsubstrate in an arrayed waveguide grating may determine in large partthe resulting birefringence. However, given a top cladding composition,a width of the waveguide core may be pre-selected that provides aminimized birefringence, i.e., a zero, or near zero, polarizationdependent wavelength. A tuned top cladding is used to describe apre-existing dopant concentration within a top cladding material.

In calculating the desired width of a waveguide core that provides thedesired level of birefringence for a given tuned top cladding having adopant in the range of 3-6% (by weight), the distribution of stress inthe top cladding may be determined over a change in temperature. Fromthis distribution of stress, a relationship may be determined betweenthe polarization dependent wavelength and various widths of thewaveguide in the arrayed waveguide grating. This relationship maydetermine a zero value, or near zero value, of polarization dependentwavelength for a given range of waveguide widths. PDW is a polarizationdependent wavelength change and the wavelength change preferably rangesfrom about 0±0.03 nm for the present invention to be optimized. Thewidth of the waveguide may be selected such that the polarizationdependent wavelength is minimized accordingly.

In one example, waveguides and methods for manufacturing waveguidesdescribed include a top cladding layer having a dopant in the range of3-6% by weight. It has been found that this particular dopant rangeprovides desirable manufacturing characteristics, for example.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a variation on a cross-section of a substrate upon which arepresentative waveguide and top cladding may be disposed.

FIG. 2 shows an example of a chart where bi-directional stresses (MPa)in the top cladding are plotted against core width (μm).

FIG. 3 shows an example of a chart where the resulting PDW (nm) isplotted against core width (μm) to obtain a core width which may resultin a PDW of zero.

FIG. 4 shows a flow chart of one variation on a method of determining awaveguide width for a given top cladding.

DETAILED DESCRIPTION

The top cladding material which usually covers the waveguides andsubstrate may determine in large part the resulting birefringence.However, given a top cladding composition, a width of the waveguide coremay be preselected that provides a desired birefringence. A top claddingwhich is “tuned” is used to describe a pre-existing dopant concentrationwithin a top cladding material. That is, the top cladding is “tuned” toprovide desired optical characteristics (such as a particular refractiveindex) but is not necessarily “tuned” to provide the lowest possiblebirefringence as described in Ojha et al. or A. Killian et al.

A representative cross-section 10 of a substrate with a waveguide andtop cladding is shown in FIG. 1. As seen in this variation,cross-section 10 may have a substrate 12, upon which waveguide 20 andtop cladding 22 may be disposed. Substrate 12 may be comprised of up toseveral layers making up the substrate 12. Layer 14 may be a layer of,e.g., undoped SiO₂ (silica), ranging in thickness from about 15-30 μm,but is preferably about 15 μm. The thickness of a layer in this exampleis taken as the distance along the y-axis; likewise, the width of alayer in this example is taken as the distance along the x-axis.

A silicon substrate 16 may be placed upon layer 14 and preferably has athickness of about 625 μm. Then layer 18 (lower cladding) may bedeposited on silicon substrate 16 or formed by oxidizing the substratein steam, for example. Layers 14 and 18 may be grown sequentially butare preferably grown simultaneously. Layer 18, like layer 14 may becomprised of, e.g., undoped SiO₂ (silica), and may likewise have athickness ranging from about 15-30 μm, but is also preferably about 15μm. In other examples, the substrate may include a glass or quartzsubstrate.

A core layer may then be deposited on top of layer 18 using, e.g.,standard silica deposition techniques such as flame hydrolysis orplasma-enhanced chemical vapor deposition (PECVD). This core materialmay be comprised of, e.g., silica with a dopant such as Germanium and/orPhosphorus, and may preferably have a refractive index about 0.5-1%higher than the lower cladding 18 index. Alternatively, the corematerial may comprise a variety of polymer materials such as opticalgrades of polyacrylates, polymethacrylates, polysilicone, polyimide,epoxy, polyurethane, polyolefin, polycarbonate, polyamides, polyesters,etc., as well as a various copolymers thereof, such asacrylate-methacrylates, acrylic-silicones, epoxy-urethanes,amide-imides, etc. The core layer may range in thickness, h, from about5-8 μm, but is preferably about 6 μm thick. The core layer may bepatterned using, e.g., photolithography and reactive ion etching, and anintermediate hard mask layer, such as chrome may be used, to define awaveguide core 20 preferably having a rectangular cross section.Waveguide core 20 also has a width, w, which may be varied accordinglyto reduce the birefringence, as described below in detail.

After core 20 is etched, a silica top cladding layer 22 may be depositedon the structure. Although top cladding 22 may be comprised of specifiedcompositions to reduce birefringence, as discussed above, this may be anunreliable method because of the difficulty in producing a top claddingcomposition which yields the required physical properties. A typical topcladding 22 may be comprised of, e.g., SiO₂ doped with Boron, but theconcentration range of such dopants may be limited. In one example, topcladding 22 is doped with, e.g., Boron, in a range of about 3-6% (wt.),which may provide adequate performance while exhibiting improvedchemical stability of the material and hence easier, more robustmanufacturing compared to dopant levels above or below the range ofabout 3-6% (wt.). For example, a concentration of Boron below about 3%may yield glass which is too viscous for manufacturing and aconcentration of Boron above about 6% may yield to phase separationduring manufacture, thereby causing the device to become morecomplicated to produce. Using a concentration of dopant between 3-6%(wt.) can be utilized or optimized by adjusting the minimum allowablegap between waveguides, adjusting the annealing schedules (e.g., addingmore incremental deposit-anneal steps to achieve top clad, or reducingthe total thickness of reflowable top-clad), properties and geometriesof core-etch, surface chemistry, diffusion rates, thermal history, etc.

As part of the present invention, a table of illustrative materialproperties, i.e., elastic modulus, E, and coefficient of thermalexpansion (CTE), are shown below in Table 1 for the purposes ofdiscussion.

TABLE 1 Example of material properties in a variation of a waveguide.Waveguide element E (GPa) CTE (×10⁻⁶ ° C.⁻¹ Top cladding 44 * Waveguidecore 62 2 Silicon (substrate) 200 3.5 Undoped SiO₂ 70 1 * Discussedbelow.

During the manufacture of an AWG, the layers of material are typicallyproduced at elevated temperatures or they may require high temperaturetreatment to ensure homogeneity. Processing temperatures may be as highas about 900° C. Upon cooling, stresses may become induced into thewaveguide core 20 because of the differing CTE values between, e.g.,layer 18 and top cladding 22, thereby resulting in undesiredbirefringence in light carried through core 20. According to theinvention, varying the width, w, of the waveguide core 20 may beselected to reduce or eliminate birefringence for a given tuned topcladding 22 having a pre-existing dopant concentration. The CTE mismatchfrom ambient temperature to a device operating temperature may typicallybe ignored for the purposes of the present invention because of theminimal effect such a temperature change may have.

In calculating an optimal width for birefringence reduction for a giventop cladding 22, a stress value may be induced within core 20 in itsblanket form. This stress value is typically due to a CTE mismatchbetween the film and the substrate and may result in a curvature withinthe plane of the substrate. This curvature may be measured and fromthere, the stress in the core film may be calculated by the followingequation (1):

$\begin{matrix}{\sigma_{core} = \frac{{E_{sub}( t_{sub} )}^{2}}{6\;{Rt}_{core}}} & (1)\end{matrix}$where,

-   σ_(core)=stress value induced in core 20 by a curvature of substrate    16;-   E_(sub)=elastic modulus of the substrate 16;-   t_(sub)=substrate thickness;-   R=radius of curvature of a given wafer from which a device may be    manufactured from according to the present invention;-   t_(core)=waveguide core 20 thickness, h.

In one variation, an example using the values as illustrated in Table 1and t_(sub)=625 μm in equation (1) may yield the following in equation(2):

$\begin{matrix}{\sigma_{core} = {\frac{( {200 \times 10^{9}} )( {625 \times 10^{- 6}} )}{6( {- 22} )( {6 \times 10^{- 6}} )} = {{- 98}\mspace{14mu}{MPa}}}} & (2)\end{matrix}$

Once calculated, the value of σ_(core) may be substituted into thefollowing equation (3) to calculate the CTE value, α_(core), of core 20:

$\begin{matrix}{\alpha_{core} = {\alpha_{sub} - \frac{\sigma_{core}}{E_{core}( {\Delta\; T} )}}} & (3)\end{matrix}$where,

-   α_(core)=calculated CTE of core 20;-   α_(sub)=CTE of substrate 16;-   E_(core)=elastic modulus of core 20;-   ΔT=temperature range through which the materials undergo.

Substituting the values from equation (2) and Table 1 may yield thefollowing equation (4):

$\begin{matrix}\begin{matrix}{\alpha_{core} = {( {3.5 \times 10^{- 6}} ) - \frac{{- 98} \times 10^{6}}{( {62 \times 10^{9}} )( {- 900} )}}} \\{= {{1.74 \times 10^{- 6}} \cong {2.0 \times 10^{- 6}{^\circ}\mspace{14mu}{C.^{- 1}}}}}\end{matrix} & (4)\end{matrix}$These calculations may be performed for each individual layer withincross-section 10 to obtain the stress and CTE values of each layer.

A stress value of top cladding 22 induced by CTE differences over thetemperature range ΔT, e.g., about 900° C. from manufacturing/processingtemperature to ambient temperature, may also be calculated. As discussedabove, if top cladding 22 were doped with a high concentration, e.g.,about 9% (wt.) of Boron, a tensile stress may be induced in top cladding22; on the other hand, if top cladding 22 were doped with a lowconcentration, e.g., about 3% (wt.) of Boron, a compressive stress orzero load may be induced in top cladding 22. In calculating the stress,the following equation (5) may be used:σ_(tc) =E _(tc)(−α_(tc)+α_(sub))(ΔT)  (5)where,

-   σ_(tc)=stress value induced in top cladding 22 by CTE differences    between top cladding 22 and substrate 16;-   α_(tc)=CTE of top cladding 22;-   α_(sub)=CTE of substrate 16;-   E_(tc)=elastic modulus of top cladding 22;-   ΔT=temperature range through which the materials undergo.

Substituting in values from Table 1 for a variation where a top cladding22 has a CTE of about, e.g., α_(tc)=3.2×10⁻⁶° C.⁻¹, and undergoes atemperature change of about, e.g., ΔT=900° C., may yield the followingequation (6). Whereas, top cladding 22 having a CTE of about, e.g.,α_(tc)=4.0×10⁻⁶° C.⁻¹, may yield the following equation (7).σ_(tc)=(44×10⁹)(−3.2+3.5)(−900)(×10⁻⁶)=−12 MPa  (6)σ_(tc)=(44×10⁹)(−4.0+3.5)(−900)(×10⁻⁶)=20 MPa  (7)

As discussed above, a difference in effective index for the TE and TMpolarization states is a reason for the occurrence of PDW, as expressedin equation (8):

$\begin{matrix}{{{PDW} = {\lambda_{c{({TM})}} - {\lambda_{c{({TE})}}\mspace{14mu}{where}}}},} & (8) \\{\lambda_{c} = {\beta\frac{\Delta\; L}{m}}} & (9)\end{matrix}$

-   λ_(c)=wavelength of light through waveguide core 20;-   β=effective refractive index;-   ΔL=distance traveled by the light through waveguide core 20;-   m=diffraction order.    Substituting equation (9) into equation (8) yields the following    result in equation (10):

$\begin{matrix}{{PDW} = {( {\beta_{TM} - \beta_{TE}} )\frac{\Delta\; L}{m}}} & (10)\end{matrix}$

The difference between β_(TM) and β_(TE) may be a result of differingstresses in the x- and y-directions along waveguide core 20. Thisdifference may vary as the width, w, of core 20 varies. As seen in FIG.2 in chart 30, stress (in MPa) induced in top cladding 22 may be plottedagainst core width (in μm) to see the relationship. Curve 32 representsthe increasing absolute value of stress, σ_(x), along the x-axis as thewidth, w, of core 20 increases. On the other hand, curve 34 shows adecreasing stress state, σ_(y) along the y-axis as the width of core 20increases. The overall stress state between σ_(x) and σ_(y) is shown incurve 36, where the stress state is shown to increase as core widthincreases.

In practice, stress calculations and stress distributions may be complexand are preferably solved through the use of computer simulations, e.g.,finite element modeling/analysis, which may account for geometry,differences in material properties, and external factors such astemperature changes and forces. Given the different CTE and stressvalues for each of the layers, as described above, the stressdistributions may be calculated over a range of varying core 20 widths,w, e.g., widths up to 11 μm and up. That is, thermal stressdistributions in waveguide core 20 and surrounding cladding 22 may becalculated for each core 20 width over a range of widths.

Once the stress distributions are calculated, as described above, anindex distribution may be calculated according to the followingequations (11) and (12):n _(y)(x,y)=n _(y0)(x,y)−c ₁σ_(y)(x,y)−c ₂[σ_(z)(x,y)+σ_(x)(x,y)]  (11)n _(x)(x,y)=n _(x0)(x,y)−c ₁σ_(x)(x,y)−c ₂└σ_(y)(x,y)+σ_(z)(x,y)┘  (12)Constants c₁ and c₂ are stress optic coefficients where, e.g.,c₁=7.56×10⁻⁷ MPa⁻¹ and c₂=4.18×10⁻⁶ MPa⁻¹; and where, e.g.,n_(xo)=n_(yo)=1.4455 for top clad 22 and n_(χo)=n_(yo)=1.455 for core20.

Properties such as refractive index can have complex distributionsthroughout a device and may be solved through computer simulation of thegeometry, refractive indices, and absorption of the waveguide andcladding, as well as index change mechanisms, e.g., temperature changes,using commonly available photonic software such as BeamPROP, made byRSoft, Inc. of Ossining, N.Y. From such software tools and from thestress results and stress distributions, the effective refractive indexmay be found. With the results of refractive index distributionaccording to equations (11) and (12), the effective refractive index βmay be calculated using, e.g., the BeamPROP simulation. This may beperformed at least once for the TE mode (n_(x) distribution) and atleast once for the TM mode (n_(y) distribution). Once the results ofβ_(TM) and β_(TE) have been calculated, PDW may finally be calculatedwith equation (10).

An example of another method of PDW reduction which varies the lengthsof the relevant waveguides as well as utilizes the results of β_(TM) andβ_(TE) may be found in the commonly-assigned U.S. patent applicationSer. No. 09/870,876, entitled “Arrayed Waveguide Grating With WaveguidesOf Unequal Widths” to Kenneth McGreer, filed on May 30, 2001 and whichis incorporated herein by reference in its entirety.

FIG. 3 illustrates the relationship in chart 40 between PDW (in nm) andcore width (in μm) for a top cladding having several different CTEvalues. Curve 42 represents a variation where a top cladding 22 has aCTE of about 4.0×10⁶° C.⁻¹ and a resulting σ_(ct) equal to about 20 MPafor a ΔT of about 900° C. Curve 46 represents another variation wheretop cladding 22 CTE is about 3.6×10⁻⁶° C.⁻¹ and a resulting σ_(tc) equalto about 4 MPa. Likewise, curve 48 represents yet another variationwhere the CTE of top cladding 22 is about 3.2×10⁻⁶° C.⁻¹ and a resultingσ_(tc) equal to about −12 MPa. As seen, as top cladding 22 CTE islowered below that of substrate 16 (where CTE is about 3.5×10⁻⁶° C.⁻¹ inthis variation), PDW generally increases. Data points 44 representexperimental results and are shown to show the close approximation andvalidity of the methods described above.

Boundary 50 represents a state where PDW is zero. Accordingly, for everytop cladding 22 process, e.g., a high Boron concentration and high CTEor a low Boron concentration and low CTE, a corresponding width, w, forwaveguide core 20, such as the width corresponding to intersection 52,may be found that results in a zero PDW state.

As described above, in calculating a width of a waveguide core thatprovides a desired birefringence for a given tuned top claddingcomposition, the distribution of stress in a top cladding over a changein temperature may be determined. From the distribution of stress, arelationship may be determined between polarization dependent wavelengthand a width of the waveguide in the arrayed waveguide grating. From thisrelationship, the width of the waveguide may be selected such that thepolarization dependent wavelength is minimized or reduced to a desiredvalue to compensate for birefringence.

A flow chart 60 is shown in FIG. 4 for one variation on a method ofdetermining waveguide width for a given top cladding. Operation 62 maybegin by determining an elastic modulus of each of the constituentlayers, e.g., layers 14 to 22. This may be accomplished by measuring themodulus directly. Operation 64 may follow where a stress value may bedetermined in each of the layers, e.g., layers 14 to 22. In operation66, the CTE values of each of the layers may be calculated by a methodas described above. Operation 68 may follow where a stress distributionmay be calculated in, e.g., core 20 and top cladding 22, for variouscore widths, w, to determine a thermal residual stress distribution duein part to CTE mismatches between different material types. Suchcalculations may be performed by computer simulations using, e.g.,finite element modeling/analysis tools.

Operation 70 may follow where the stress distribution calculated inoperation 68 may be mapped into an index distribution using aconventional stress optic coefficient. The index distribution may bedetermined by any of the methods as described above. Operation 72 mayfollow where an effective index of core 20 and top cladding 22 may bedetermined by simulations using, e.g., photonic software such asBeamPROP. From the results of operation 72, a PDW may be calculated inoperation 74 by a method as described above. Operation 76 may followwhere a relationship between PDW and various core widths, w, may bedetermined by, e.g., plotting the results on a chart as described above.An appropriate core width may accordingly be selected in which PDW isminimized, i.e., results in zero, or near zero within about 3%, PDW forany given top cladding material by utilizing the methods describedabove.

The applications of the methods and apparatus discussed above are notlimited to the numerical examples given or to the disclosed materialshaving specific CTEs, but may also include any number of furtherapplications, e.g., different material types, different CTE values, etc.Modification of the above-described methods and apparatus for carryingout the invention, and variations of aspects of the invention that areobvious to those of skill in the art are intended to be within the scopeof the claims.

1. A method of controlling birefringence in an arrayed waveguide gratingcomprising: determining a stress distribution in a top cladding layerand in at least one waveguide in the arrayed waveguide grating over achange in manufacturing temperature, the top cladding layer having adopant in the range of 3-6% by weight; determining a relationshipbetween polarization dependent wavelength and a width of the waveguidefrom the stress distribution; and selecting the width of the waveguidesuch that the polarization dependent wavelength is a predeterminedvalue.
 2. The method of claim 1 wherein the predetermined value is aminimized polarization dependent wavelength.
 3. The method of claim 1further comprising determining an elastic modulus of each of the topcladding layer and the waveguide prior to determining the stressdistribution.
 4. The method of claim 3 further comprising determining acoefficient of thermal expansion in each of the top cladding layer andthe waveguide.
 5. The method of claim 1 wherein determining therelationship between polarization dependent wavelength and the waveguidewidth from the stress distribution further comprises mapping the stressdistribution into an index distribution.
 6. The method of claim 5further comprising determining an effective index of each of thewaveguide and the top cladding.
 7. The method of claim 1 whereindetermining the relationship between polarization dependent wavelengthand the waveguide width from the stress distribution further comprisesdetermining a distribution of refractive index in the top cladding fromthe stress distribution.
 8. The method of claim 2 wherein the minimizedpolarization dependent wavelength is zero.
 9. The method of claim 1wherein the top cladding layer is disposed over the waveguide.
 10. Themethod of claim 9 wherein the top cladding layer is disposed over asubstrate.
 11. The method of claim 10 wherein the substrate comprisessilicon.
 12. The method of claim 10 wherein the substrate has athickness of about 625 μm.
 13. The method of claim 10 wherein thesubstrate is disposed over a first layer of SiO₂.
 14. The method ofclaim 13 wherein a second layer of SiO₂ is disposed over the substrate.15. The method of claim 1 wherein the waveguide further comprises atleast one dopant.
 16. The method of claim 15 wherein the at least onedopant is selected from the group consisting of Germanium andPhosphorus.
 17. The method of claim 1 wherein the waveguide comprises atleast one material selected from the group consisting of polyacrylates,polymethacrylates, polysilicone, polyimide, epoxy, polyurethane,polyolefin, polycarbonate, polyamides, polyesters, acrylate-methacrylatecopolymers, acrylic-silicone copolymers, epoxy-urethane copolymers, andamide-imide copolymers.
 18. The method of claim 1 wherein the dopantcomprises Boron.
 19. The method of claim 1 wherein the change inmanufacturing temperature is about 900° C.
 20. An arrayed waveguidegrating manufactured according to the method of claim 1.